BOUNDARY RIGIDITY AND FILLING VOLUME
MINIMALITY OF METRICS CLOSE TO A FLAT ONE.
D. BURAGO AND S. IVANOV
1. Introduction
Let (M n , g) be a compact Riemannian manifold with boundary ∂M . Its
boundary distance function is the restriction of the Riemannian distance
dg to ∂M × ∂M . The term “boundary rigidity” means that the metric is
uniquely determined by its boundary distance function. More precisely,
Definition 1.1. (M, g) is boundary rigid if every compact Riemannian manifold (M̃ , g̃) with the same boundary and the same boundary distance function is isometric to (M, g) via a boundary preserving isometry.
It is easy to construct metrics that are not boundary rigid. For example,
consider a metric on a disc with a “big bump” around a point p, such
that the distance from p to the boundary is greater than the diameter of
the boundary. Since no minimal geodesic between boundary points passes
through p, a perturbation of the metric near p does not change the boundary
distance function.
Thus one has to impose restrictions on the metric in order to make the
boundary rigidity problem sensible. One natural restriction is the following:
a Riemannian manifold (M, g) is called simple if the boundary ∂M is strictly
convex, every two points x, y ∈ M are connected by a unique geodesic, and
geodesics have no conjugate points (cf. [15]). A more general condition
called SGM (“strong geodesic minimizing”) was introduced in [8] in order
to allow non-convex boundaries. Note that if (M, g) is simple, then M is a
topological disc. The simplicity of (M, g) can be seen from the boundary
distance function. The convexity of ∂M is equivalent to a (local) inequality
between boundary distances and intrinsic distances of ∂M . The uniquieness
of geodesics is equivalent to smoothness of the boundary distances. Thus if
two Riemannian manifolds have the same boundary and the same boundary
distance functions, then either both are simple or both are not.
Conjecture 1.2 (Michel [15]). All simple manifolds are boundary rigid.
Pestov and Uhlmann [16] proved this conjecture in dimension 2. In higher
dimensions, few examples of boundary rigid metrics are known. They are:
regions in Rn [11], in the open hemisphere [15], and in symmetric spaces of
The first author is partially supported by the NSF Grant DMS-0412166. The second
author is partially supported by the RFBR grant 05-01-00939.
1
negative curvature (follows from the main result of [7]). We refer the reader
to [9] and [16] for a survey of boundary rigidity, other inverse problems, and
their applications.
One of the main results of this paper asserts that if (M, g) is C 2 -close to
a region in the Euclidean space, then (M, g) is rigid. For instance, to the
best of our knowledge, this is the first known example of boundary rigid
metrics in higher dimensions which are not locally-symmetric. Our result
also requires only C 2 -smoothness, so even in dimension 2 it is not completely
covered by Pestov-Uhlmann’s 2-dimensional theorem [16].
Our approach to boundary rigidity grew from [5] and [6], where we study
minimality of flats in normed spaces, asymptotic volume of Finsler tori, and
ellipticity of surface area functionals. Even though our proof is not directly
based on Finsler geometry, it is strongly motivated by Finsler considerations.
Boundary rigidity here is treated as the equality case of the minimal filling
problem discussed in [5] and [13].
Definition 1.3. (M, g) is a minimal filling if, for every compact (M̃ , g̃) with
∂ M̃ = ∂M , the inequality
dg̃ (x, y) ≥ dg (x, y)
for all x, y ∈ ∂M
implies
vol(M̃ , g̃) ≥ vol(M, g).
We say that (M, g) is a minimal orientable filling if the same holds under
the additional assumption that (M̃ , g̃) is orientable.
Conjecture 1.4. Every simple manifold is a minimal filling.
If (M, g) is simple, then vol(M, g) is uniquely determined by dg , namely
there is an integral formula expressing vol(M, g) via dg and its first order
derivatives (the Santaló formula, [17]). It is not clear though whether the
formula is monotone in dg .
Our approach to Michel’s Conjecture is to prove Conjecture 1.4 and then
to obtain Michel’s Conjecture by studying the equality case. So far we were
able to carry out this plan for metrics close to a Euclidean one.
The main result of this paper is the following Theorem:
Theorem 1. Let M ⊂ Rn be a compact region with a smooth boundary.
There exists a C 2 -neighborhood U of the Euclidean metric gE on M such
that, every g ∈ U is a minimal orientable filling and boundary rigid.
One can check that actually we show that there is a c(n) > 0 such that,
if g is a Riemannian metric in Rn satisfying g = gE outside BR (0) and
, then for any Ω ⊂ BR (0), the space (Ω, g) is a minimal orientable
|Kσ | < c(n)
R2
filling and boundary rigid.
We do not know if the orientability assumption can be removed; this
seems to be a rather intriguing question.
2
Known higher-dimensional examples of minimal fillings form a subset of
known examples of rigid metrics: regions in Rn (follows from the Besikovitch
inequality [2]) and regions in symmetric spaces of negative curvature [7].
There are many more examples of locally rigid metrics: for instance, simple almost nonpositively curved metrics and simple analytic metrics are locally rigid [10, 18]. The manifold (M, g) is said to be locally (boundary) rigid
if every compact Riemannian manifold (M̃ , g̃) with the same boundary and
the same boundary distance function is isometric to (M, g) via a boundary
preserving isometry provided that g and g̃ are a priori sufficiently close. We
want to emphasize that in Theorem 1 we do not impose any restrictions on
M̃ .
All 2-dimensional simple manifolds are minimal fillings in a restricted
sense: they are minimal only within the class of fillings homeomorphic to
the disc [13]. In general (when M̃ from definition 1.3 may have handles), it
is not known even if the standard hemisphere is a minimal orientable filling.
That is, the filling volume (in the sense of M. Gromov) of the standard circle
is not known.
However, it has been noticed by M. Gromov [11] that if n ≥ 3, then one
can assume that M̃ ' Dn without loss of generality (i.e., the orientbale
filling volume can be realized by topological discs).
Remark 1.5. The Finsler case was very important for motivating our argument. Little is known about minimality of Finsler metrics, even though the
Santaló formula still yields the normalized symplectic volume of the unit
cotangent bundle (the Holmes–Thompson volume). This work originated
from our study of minimality of flat Finsler metrics. However, there is no
rigidity in the Finsler case. Here is a simple example.
Example: Let (M, g) be a simple Riemannian manifold. For every p ∈
∂M define a function fp : M → R by
fp (x) = distg (p, x).
Let {f˜p } be a C 3 perturbation of {fp } in the interior of M . Then {f˜p } is a
family of distance functions of a Finsler metric with the same boundary distances (this metric is possibly non-symmetric, but it can be made symmetric
with some additional work)
This Finsler metric is defined by
kvkx = sup{dfp (v)},
x ∈ M, v ∈ Tx M.
p
We obtain Theorem 1 as a corollary of the following (more technical and
more general):
Theorem 2. Let M ⊂ Rn be a compact region with a smooth boundary.
There exists a C 2 -neighborhood U of the Euclidean metric gE on M such
that for every g ∈ U the following holds.
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If (M̃ , g̃) is an orientable piecewise C 0 Riemannian manifold such that
∂ M̃ = ∂M and the respective Riemannian distance functions d and d˜ satisfy
˜ y) ≥ d(x, y)
d(x,
for all x, y ∈ ∂M ,
then
1. vol(M̃ , g̃) ≥ vol(M, g);
2. if vol(M̃ , g̃) = vol(M, g) then (M̃ , g̃) is isometric to (M, g) via a
boundary preserving isometry.
Here by a piecewise C 0 Riemannian manifold we mean a smooth manifold,
possibly with boundary, triangulated into simplices such that each simplex
is C 1 -diffeomorphic to the standard one and equipped with a continuous
Riemannian structure. The Riemannian metrics on simplices do not have
to agree on their common faces.
Deducing Theorem 1 from Theorem 2. To deduce Theorem 1 from Theorem 2 it suffices to check the following two facts.
˜ y) = d(x, y) for all x, y ∈ ∂M implies vol(M̃ , g̃) =
1. The equality d(x,
vol(M, g). Indeed, if M is convex (and hence simple), this immediately
follows from the Santaló formula. Since we do not assume convexity, M
may fail to be simple. However, it is easy to check that it still satisfies the
SGM (Strong Geodesic Minimizing) condition introduced by C. Croke [8].
Then Lemma 5.1 from [8] implies the desired equality vol(M̃ , g̃) = vol(M, g).
˜ y) = d(x, y) for all x, y ∈ ∂M also implies that M̃
2. The equality d(x,
is orientable. In fact, M̃ is homeomorphic to M . Again, if M is convex,
it is easy to show that both M and M̃ are homeomorphic to a disc. For a
general region M ⊂ RN satisfying the conditions of Theorem 1 this is the
contents of Remark 5.2 in the above mentioned paper [8].
Acknowledgements. The authors are grateful to Y. Kurylev and G. Uhlmann
for introducing them to and giving them a taste of Inverse Problems and
for many useful remarks. The first author is grateful to the organizers of
the conference “Perspectives in Inverse Problems”, 2004, Helsinki. To a
large extent, the very idea to undertake this project has been ignited by
this conference. The authors are grateful to the anonymous referee for an
extraordinary detailed and useful report and, in particular, for pointing out
a few inaccuracies in the preliminary version of the paper.
2. Plan of the Proof
In the “ideal world”, the proof of boundary rigidity should go as follows:
It is well-known that every compact metric space X can be embedded
into L∞ (X) isometrically by sending x to d(x, ·). By attaching appropriate
collars, one can assume that both boundaries ∂M = ∂ M̃ = S, where S
is a standard sphere in Rn , and that both metrics d and d˜ are extended
by the standard Euclidean metric to the outside of S. Denote by Tα S the
supporting hyperplane to S at α ∈ S. One can see that since (M, g) is
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simple, the map φ from M to L = L∞ (S) sending x to φx : S → R :
φx (α) = d(x, Tp S) is also an isometry (in the strongest possible sense: it
is a distance preserving map). Thus it is very tempting to think of this
embedding as a “minimal surface” in L. Applying the same construction to
M̃ one gets a Lipschitz-1 (and hence an area non-increasing) map φ̃. Since
M and M̃ have the same boundary distance function, the embeddings φ and
φ̃ coincide on the common boundary S = ∂M = ∂ M̃ . Furthermore, if d is
a flat metric, then φ is a linear embedding. Hence our assumption that d
is close to a Euclidean metric tells us that φ is close to a linear embedding.
Then all we would need to conclude the “proof” is an infinite-dimensional
analog of a well-known theorem (for instance, see Theorem 3 and Remark 3.1
of [14]) that a minimal surface close to an affine plane of the same dimension
is the unique area-minimizer among all surfaces with the same boundary.
However, this approach encounters a number of difficulties:
1. When we speak about minimal surfaces, we need to define surface
area. This is a major question. The space L naturally carries the structure
of a normed space, and there are many different notions of surface area in
normed spaces. It is very convenient to work with symplectic (the Holmes–
Thompson, [12, 19]) surface area; however, there are too many minimal
surfaces with respect to this surfaces area. We will fix this by introducing a
surface area induced by a family of L2 -structure on L.
2. We need to prove that φ is indeed a minimal surface. The fact that
it is totally geodesic does not imply by itself minimality for non-standard
surface areas (e.g., see [1]). We verify minimality by means of a rather
straightforward but cumbersome computation.
3. We need a very “robust” argument for the uniqueness of minimal
surfaces close to affine planes. Our proof models a co-dimension one argument showing that two co-dimension one minimal surfaces with the same
boundary coincide provided that both of them are graphs of functions (with
respect to the same coordinates). Indeed, if the surfaces are graphs of f
and g, consider a function v(t) = area(Graph(tf + (1 − t)g). We have
v 0 (0) = v 0 (1) = 0 by minimality of f and g. By the Cauchy inequality v is
convex on t ∈ [0, 1]. Furthermore, it is strictly convex unless f = g, and this
implies that f = g. We will generalize this argument to higher co-dimensions
(using the assumption that one of the surfaces is close to a plane).
3. Attaching a collar
This is a purely technical section. Its purpose is to reduce the problem to
a special case when M is a Euclidean disc of radius 1, and g coincides with
1
.
the standard Euclidean metric outside the ball of radius 10n
Proposition 3.1. Theorem 2 follows from its special case when
(i) M is a unit disc D = B1 (0) ⊂ Rn and g coincides with the standard
Euclidean metric gE on the “collar” N = B1 (0) \ B1/10n (0);
(ii) M̃ contains N (with ∂ M̃ = ∂N ) and g̃ = g on N ;
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(iii) the distance functions dg and dg̃ satisfy the inequality dg̃ (x, y) ≥
dg (x, y) for all x, y ∈ N .
Proof. Let (M, g) and (M̃ , g̃) be as in Theorem 2. By means of re-scaling
we assume that M is contained in the ball B1/20n (0) ⊂ Rn . We extend g
to a smooth metric on the whole Rn so that g remains C 2 -close to gE and
g = gE outside the ball B1/10n (0). (The extended metric is denoted by the
same letter g.)
Let M + = (D, g). We can think of M + as the result of attaching another
“collar” N 0 = D \ M to M . Attaching the same collar (N 0 , g) to (M̃ , g̃) we
obtain a manifold M̃ + = M̃ ∪ N 0 with a piecewise C 0 Riemannian metric
(which we will also denote by g̃). Note that N ⊂ N 0 , so g̃ = g = gE on N .
The new spaces (M + , g) and (M̃ + , g̃) satisfy the conditions (i)–(iii). The
first two are obvious. To verify (iii), consider x, y ∈ N and observe that
the length distance d(M̃ + ,g̃) (x, y) depends only on g|N and d(M̃ ,g̃) |∂M ×∂M
and the latter dependency is monotonous. Since d(M̃ ,g̃) ≥ d(M,g) on ∂M , it
follows that d(M̃ + ,g̃) (x, y) ≥ d(M + ,g) (x, y).
It remains to note that the conclusion of Theorem 2 for (M + , g) and
(M̃ + , g̃) implies that for (M, g) and (M, g̃).
Convention. From now we assume that (M, g) and (M̃ , g̃) from Theorem 2 satisfy the additional assumptions from Proposition 3.1.
4. Distance-preserving embedding into L∞
We fix the following notation: S = ∂M = ∂ M̃ = S n−1 (recall that M = D
by the convention from the previous section); L = L∞ (S).
The goal of this section is to construct Lipschitz-1 maps ΦE , Φ and Φ̃
from (Rn , gE ), (M, g) and (M̃ , g̃) resp., to L. When we speak about maps
to L, we always keep in mind the following construction.
Definition 4.1. Given a (measurable) family {Fα }α∈S , of uniformly locally
bounded functions Fα : M → R, one can think of this family as a map
F : M → L where F (x)(α) = Fα (x) for x ∈ M , α ∈ S. We say that Fα are
coordinate functions of F .
Note that a family {Fα } defining a given map F is not unique and may
be defined only for almost every α.
Lemma 4.2. If F : M → L is defined by a family {Fα } of coordinate
functions and every Fα is Lipschitz-1, then so is F .
Proof. Immediately from the definition of the distance in L = L∞ (S).
Conversely, every Lipschitz-1 map Φ : M → L can be represented by
Lipschitz-1 coordinate functions. We prove this in the next section, cf. Lemma
5.1.
6
Definition 4.3. Define ΦE : Rn → L by
ΦE (x)(α) = hx, αi,
x ∈ Rn , α ∈ S
where h, i is the standard scalar product in Rn .
Obviously ΦE is a linear map. For α ∈ S, the corresponding coordinate
function ΦEα : Rn → R is the scalar multiplication by α. Since α is a unit
vector (recall that S = ∂D is the unit sphere in Rn ), ΦEα is a Lipschitz-1
function. Then so is ΦE . Moreover ΦE is an isometric embedding. Indeed,
kΦE (x)k = suphx, αi = |x|.
α∈S
Definition 4.4. Let Φ : M → L be a map whose coordinate functions
{Φα }α∈S are given by
Φα (x) = 1 − distg (x, Hα )
where Hα is the hyperplane tangent to S at α, and distg is the distance with
respect to g (assuming that g = gE outside M ; recall that this is a smooth
extension).
Observe that if this definition is applied to the Euclidean metric gE in
place of g, it yields the map ΦE |M . Indeed, the Euclidean distance from Hα
to x ∈ M equals 1 − hx, αi.
Since the metric g is C 2 -close to gE , the hyperplanes Hα have no focal
points in M , hence the functions Φα are smooth distance-like functions.
The Riemannian gradient of Φα at x ∈ M is the initial velocity vector of
the unique minimal geodesic connecting x to Hα .
Definition 4.5. Define a map G : M × S → U T M by
G(x, α) = grad Φα (x)
where the gradient is taken with respect to the metric g.
We denote by GE the similar function for gE in place of g. Then
gE (x, α) = (x, α) ∈ Rn × S ∼
= U T Rn
(recall that S is the unit sphere in Rn ).
Proposition 4.6. 1. Φ : (M, g) → L is a distance preserving map.
2. Φ is C 1 smooth.
3. The map G : M × S → U T M is a diffeomorphism.
4. Φ is C 1 -close to ΦE ; G is C 1 -close to GE .
Proof. 1. Every Φα is Lipschitz-1, so is Φ (by Lemma 4.2). It remains
to show that kΦ(x) − Φ(y)k ≥ dg (x, y), for all x, y ∈ M . Since Φα (x) is
continuous in α, we have
kΦ(x) − Φ(y)k = sup |Φα (x) − Φα (y)|
α∈S
Let γ be a geodesic from x through y (x = γ(0), y = γ(t1 )). It is close
to a straight line while in M and coincides with a straight line after it
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leaves M . Eventually γ hits orthogonally one of the hyperplanes Hα , that
is, γ(t2 ) ∈ Hα and γ 0 (t2 ) ⊥ Hα for some α ∈ S and t2 > t1 . Since Hα has
no focal points in M , we have distg (x, Hα ) = t2 and distg (y, Hα ) = t2 − t1 .
Then
|Φα (x) − Φα (y)| = | distg (x, Hα ) − distg (y, Hα )| = t1 = dg (x, y)
and the desired inequality follows.
2–4. Since g is C 2 -close to gE , the geodesic flow of g is C 1 -close to that
of gE . In particular, the hyperplanes have no focal points in M . Then the
distance functions of the hyperplanes and their gradients are recovered from
the union of the hyperplanes’ normal geodesic flows via the implicit function
theorem, and they are C 1 close to their Euclidean counterparts.
Remark 4.7. The assumption that g is close to gE is needed only for the
last statement of the proposition. The first three would follow for any simple
metric g if we defined Φα (x) = distg (x, α).
Now we are in a position to define a “surface” Φ̃ : M̃ → L spanning the
same boundary as Φ. All we need is a Lipschitz-1 extension Φ̃α of Φα |∂M
from ∂M = ∂ M̃ to M̃ . Such an extension exists due to the fact that Φα |∂ M̃
is Lipschitz-1 w.r.t. dg̃ . Indeed, it is Lipschitz-1 w.r.t. dg and dg̃ ≥ dg on ∂ M̃ .
(This is the only point where we use this key assumption of Theorem 2.) In
order to ensure that the family {Φ̃α } is measurable (in fact, continuous), we
define an extension by an explicit formula. We also want Φ̃ to be reasonably
close to Φ, so we cut off too large values of the functions.
Definition 4.8. Let Φ̃ : M̃ → L be a map whose coordinate functions
{Φ̃α }α∈S are given by
2
Φ̃α (x) = cutoff inf {Φα (y) + dg̃ (x, y)},
+ distg̃ (x, M̃ \ N )
y∈N
10n
where
cutoff(a, b) = min{b, max{−b, a}}.
Recall that N is the “collar”, cf. Proposition 3.1.
Proposition 4.9. 1. Φ̃ : (M̃ , g̃) → L is a Lipschitz-1 map.
2. Φ|N = Φ̃|N .
2
centered at the origin
3. Φ̃(M̃ \ N ) is contained in the ball of radius 10n
of L.
Proof. 1. Every Φ̃α is Lipschitz-1 since it is obtained from a family of
Lipschitz-1 functions by means of suprema and infima. Then by Lemma 4.2
Φ̃ is Lipschitz-1.
2. Since Φ is close to a linear isometry ΦE and M \ N is the disc of radius
1
2
10n , we have supM \N |Φα | ≤ 10n . Let x ∈ N . Then
|Φα (x)| ≤ sup |Φα | + distg (x, M̃ \ N ) ≤
M \N
8
2
+ distg̃ (x, M̃ \ N ),
10n
hence the cutoff does not apply. Furthermore,
Φα (x) ≤ Φα (y) + dg (x, y) ≤ Φα (y) + dg̃ (x, y)
for all y ∈ N . (The inequalities follow from the facts that Φα is Lipschitz-1
w.r.t. g and dg ≤ dg̃ on N .) Then the infimum in the definition of Φ̃α is
attained at y = x and Φ̃α (x) = Φα (x).
2
2
3. If x ∈ M̃ \N , then |Φα (x)| ≤ 10n
due to cutoff, hence kΦ̃(x)k ≤ 10n
. 5. Coordinates and derivatives
This section is technical. Its purpose is to validate our view of L as a
“coordinate space” and Φ̃ as a “surface” (with tangent planes) in this space.
In this section M denotes an arbitrary Riemannian manifold while S =
S n−1 and L = L∞ (S) are the same as in the previous section. Recall that a
family {Fα } of functions on M defines a map F : M → L (cf. Definition 4.1).
The converse is more complicated since a point in L is a “function defined
a.e.” whose individual values do not make sense.
Lemma 5.1. 1. Every Lipschitz map F : M → L can be represented by
a family {Fα }α∈S of coordinate functions so that every Fα : M → R is
Lipschitz with the same Lipschitz constant.
2. If {Fα } and {Fα0 } are two such representations, then for almost every
α ∈ S, Fα = Fα0 everywhere on M .
3. If, in addition, M is a vector space and F is linear, then Fα is linear
for almost every α.
Proof. 1. Let X be a countable dense subset of M . For every x ∈ X, pick a
function fx : S → R representing F (x) ∈ L∞ (S). Then for every x, y ∈ X,
|fx (α) − fy (α)| ≤ C|xy| for a.e. α ∈ S
where C is the Lipschitz constant of F and |xy| is the distance in M . Since X
is countable, we can redefine fx (α) to be zero whenever the above inequality
fails for at least one y ∈ X. Then |fx (α) − fy (α)| ≤ C|xy| for all x, y ∈ X
and α ∈ S, and we get a family of Lipschitz functions Fα : X → M . Every
Fα admits a unique Lipschitz extension to the whole M , also denoted by Fα .
It remains to note that for every z ∈ M , the function α 7→ Fα (z) represents
F (z) in L∞ (S). Indeed, if fz : S → R represents F (z), then for almost
every α the inequality |fz (α) − fx (α)| ≤ C|zx| holds for all x ∈ X, and this
property uniquely determines fz (α) = Fα (z).
2. For every x ∈ M , we have Fα (x) = Fα0 (x) for almost all α. Then by
Fubini, for almost every α, the relation Fα (x) = Fα0 (x) holds for almost all
x ∈ M , and hence for all x ∈ M by continuity of Fα and Fα0 .
3. Similarly, for almost every α, the relation Fα (x + y) = Fα (x) + Fα (y)
holds for almost all pairs (x, y), and hence for all x, y.
Definition 5.2. We say that a Lipschitz map F : M → L is weakly differentiable at x ∈ M if the coordinate function Fα is differentiable at x for
9
almost every α. If so, we define the derivative dx F : Tx M → L to be the
map whose coordinate functions are dx Fα .
We need the following version of Rademacher’s Theorem:
Lemma 5.3. Let F : M → L be a Lipschitz function. Then
1. F is weakly differentiable almost everywhere;
2. If F is weakly differentiable at x ∈ M , then the derivative dx F :
Tx M → L is a Lipschitz linear map with the same Lipschitz constant.
Proof. Every coordinate function Fα is Lipschitz and hence differentiable a.e.
(by Rademacher’s Theorem). Then by Fubini almost every x ∈ M satisfies
the following: for almost all α, Fα is differentiable at x. Furthermore,
kdx Fα k ≤ C where C is a Lipschitz constant for F . Then lemmas 4.2 and
5.1, imply that dx F : Tx M → L is correctly defined and Lipschitz with the
same constant.
The map dx F introduced above is not a derivative in any traditional sense.
We will use only a limited set of features of this “derivative”, namely the
following chain rule.
Lemma 5.4. Let F : M → L be a Lipschitz function weakly differentiable
at x ∈ M , and let µ be a continuous finite measure on S (that is, a measure
with an L1 density). Then
1. If L : L → R is a linear function of the form
Z
L(f ) =
f dµ
S
then L ◦ F is differentiable at x and
dx (L ◦ F ) = L ◦ dx F.
2. If W is a finite-dimensional subspace of L and P : L → W is the
orthogonal projection w.r.t. the L2 structure defined by µ, then P ◦ F is
differentiable at x and
dx (L ◦ P ) = L ◦ dx P.
Proof. 1. Since the functions Fα are uniformly Lipschitz, the lemma follows
immediately by differentiation under the symbol of integration.
2. The first part of lemma implies that for every w ∈ W , the function
f 7→ hf, wi on L commutes with differentiation. Applying this to every w
from a basis of W yields the second part.
6. A Riemannian structure on L
Definition 6.1. Let µ be a probability measure on S. We define a scalar
product h, iµ on L by
Z
hf, giµ = n f g dµ.
S
10
We denote the space L equipped with this scalar product by Lµ , and the
identical map idL regarded as a map from L to Lµ by iµ . Obviously iµ is a
Lipschitz map with Lipschitz constant n.
The normalizing factor n in the definition is introduced for the following
reason: the integral of the square of a linear function of norm one against
the normalized surface area over the unit sphere is equal to n1 .
Lemma 6.2. Let A : Rn → L be a Lipschitz-1 linear map. Then the
composition iµ ◦ A : Rn → Lµ is area non-expanding. Furthermore, if iµ ◦ A
is an area-preserving map then A and iµ ◦ A are linear isometries.
Proof. Let {Aα }α∈S be the coordinate functions of A and gµ = A∗ (h, iµ ) be
the pull-back of the scalar product in Lµ . Then
Z
gµ (v, v) = n Aα (v)2 dµ(α).
S
Hence
Z
trace(gµ ) = n
trace(A2α )dµ(α) ≤ n
S
since traceA2α = kAα k2 ≤ 1. Since gµ is a positive definite symmetric matrix,
we conclude the proof of the inequality by applying the inequality
1
det(gµ ) ≤ ( trace(gµ ))n/2 .
n
The equality case obviously follows from the equality case in the above
inequality.
Recall that we have a diffeomorphism G : M × S → U T M with G(x, α) ∈
U Tx M (cf. Definition 4.5 and Proposition 4.6). Then for every x ∈ M , the
map G(x, ·) : S → U Tx M is a diffeomorphism.
Definition 6.3. Let x ∈ M . We denote the inverse of G(x, ·) by ωx , that
is, define a map ωx : U Tx M → S by
ωx (G(x, α)) = α
for all α ∈ S.
Let µx be the push-forward by ωx of the normalized standard (n − 1)volume on the unit sphere U Tx M . For brevity, we denote Lµx by Lx and
similarly iµx by ix .
Lemma 6.4. In the above notation, ix ◦ dx Φ : Tx M → Lx is a linear
isometric embedding for every x ∈ M .
Proof. Denote U = U Tx M . For every v ∈ U ,
Z
kdx Φ(v)k2Lx = n |dx Φα (v)|2 dµx (α)
ZS
Z
= n hv, ωx−1 (α)i2 dµx (α) = n hv, ui2 du = 1
S
U
11
where du denotes the normalized (n − 1)-volume on U . The second equality
follows from the definitions of G and ωx : grad Φα (x) = G(x, α) = ωx−1 (α).
The last integral equals n1 since it does not depend on v ∈ U (due to the
symmetry of the measure), and if v ranges over an orthonormal basis of Tx M ,
the sum of the corresponding functions under the integral is the constant 1.
Recall that our surface Φ(M ) is close to an n-dimensional linear subspace
ΦE (Rn ). We want to think of this surface as a graph of a map from this
subspace to its “orthogonal complement” denoted by Q (see below). Then
we extend our family of scalar products {h, ix }x∈M to a Riemannian structure on the whole L. This Riemannian structure equals h, ix at Φ(x) and
is constant along subspaces parallel to Q. Then lemmas 6.4 and 6.2 imply
that Φ is an isometric embedding and Φ̃ is area-nonexpanding with respect
to this Riemannian structure. We are going to prove the main theorem by
comparing the areas of surfaces Φ(M ) and Φ̃(M̃ ) in the resulting infinitedimensional Riemannian space.
To avoid unnecessary technical details, we do not refer directly to the
Riemannian structure in L. Instead, we consider a projection of M̃ to M
corresponding to the projection of Φ̃(M̃ ) to Φ(M ) along Q, and define “areas” in terms of scalar products h, ix .
Definition 6.5. Let H be a Euclidean space (not necessarily finite dimensional) and ε > 0. We say that linear subspaces W1 and W2 of H are
ε-orthogonal if ∠(w1 , w2 ) ≥ π2 − ε for all nonzero vectors w1 ∈ W1 , w2 ∈ W2 .
Proposition 6.6. There is a codimension n linear subspace Q ⊂ L and a
Lipschitz map π : M̃ → M satisfying the following.
1. For every x ∈ M , Q is ε-orthogonal to the image of dx Φ in Lx for a
small ε > 0.
2. For every x ∈ M̃ , Φ(π(x)) − Φ̃(x) ∈ Q.
3. If Φ̃ is weakly differentiable at an x ∈ M̃ , then π is differentiable at x
and dx (Φ ◦ π − Φ̃)(v) ∈ Q for all v ∈ Tx M̃ .
Proof. If M is Euclidean (that is, g = gE ) then µx is independent of x
and coincides with the standard normalized (n − 1)-volume ν on S. Since
the map G is close to its Euclidean counterpart (cf. Proposition 4.6), the
measures µx are absolutely continuous with respect to ν and have densities
close to one. Thus every scalar product h, ix , x ∈ M , is close to the “flat”
L2 structure h, iν .
Let Q be the orthogonal complement to W = ΦE (Rn ) with respect to
h, iν . Since every scalar product h, ix is close to h, iν , the first assertion of
the proposition follows. Let P : L → W be the orthogonal projection with
respect to h, iν . Since Φ is C 1 close to ΦE , the map P ◦Φ is a diffeomorphism
of M to a region Ω ⊂ W , and Ω is close to the unit ball in W .
Recall that (by Proposition 4.9) Φ̃ coincides with Φ on the “collar” N ,
2
and Φ̃(M̃ \ N ) is contained within the ball of radius 10n
in L, and hence
12
within the ball of radius
define π : M̃ → M by
2
10
in Lν . Therefore P ◦ Φ̃(M̃ ) ⊂ Ω, and we can
π = (P ◦ Φ)−1 ◦ (P ◦ Φ̃).
The second assertion of the proposition follows immediately. If Φ̃ is weakly
differentiable at x, then by the second part of Lemma 5.4 the map P ◦ Φ̃ is
differentiable at x and dx (P ◦ Φ̃) = P ◦ dx Φ̃. Then the last assertion follows
since P ◦ Φ is a diffeomorphism and Φ is smooth.
Notation 6.7. We fix the notation π introduced in Proposition 6.6 for the
rest of the paper. Also introduce Φπ = Φ ◦ π and V = Φ̃ − Φπ .
Definition 6.8. If Φ̃ is weakly differentiable at an x ∈ M̃ , denote by Jx Φ̃
the Jacobian (that is, the area-expansion coefficient) of dx Φ̃ as a map from
Tx M̃ to Lπ(x) . By Lemma 5.3, Jx Φ̃ is defined for a.e. x ∈ M̃ . Then define
Z
Jx Φ̃ dx
Area(Φ̃) =
M̃
where the integral is taken with respect to the Riemannian volume on (M̃ , g̃).
Now Lemma 6.2 implies
Lemma 6.9. Area(Φ̃) ≤ vol(M̃ , g̃). The equality in this inequality implies
that Jx Φ̃ = 1 for a.e. x ∈ M̃ and dx Φ̃ is a linear isometry.
7. First Variation of Surface Area
Φπ
The maps
and Φ̃ can be connected by a linear family of maps {Φt }t∈[0,1]
from M̃ to L defined by Φt = Φπ + tV. We think of V as a vector field of
variation of a surface Φπ and introduce a quantity δA(Φπ , V) which we call
the first variation of surface area.
Definition 7.1. Let H be a (possibly infinite-dimensional) Euclidean space,
and W an oriented n-dimensional linear subspace of H. Let PW denote the
orthogonal projection to W .
For an oriented Euclidean n-space X and a linear map L : X → H, let
JW (L) denote the Jacobian determinant of PW ◦L (which takes into account
the orientation of X and W ). We also think of JW (L) as an element of Λn X ∗
(i.e., an exterior n-form on X), using the natural identification Λn X ∗ = R.
In this interpretation, JW (L) does not depend on the Euclidean structure
of X.
For linear maps L, V : X → H introduce
d δJW (L, V ) = JW (L + tV ).
dt t=0
Now define
Z
(1)
δA(Φπ , V) =
δJWπ(x) (dx Φπ , dx V) dx
M̃
13
where Wπ(x) = dπ(x) Φ(Tπ(x) M ) is the tangent space to Φ(M ) at Φπ (x)
regarded as a subspace of Lπ(x) , so the term JWπ(x) is computed with respect
to the scalar product h, iπ(x) . The quantity δA(Φπ , V) is well-defined since
both dx Φπ and dx V are defined a. e. The orientation of Wπ(x) is defined so
that the map dπ(x) Φ : Tπ(x) M → Wπ(x) is orientation-preserving.
The formula (1) can be read in two equivalent ways. First, it is an integral
of a real-valued function function against the Riemannian volume dx on
M̃ . Second, the integrand can be regarded as an exterior n-form on Tx M
(independent of the Riemannian structure), thus defining a (measurable)
differential n-form on M̃ , and δA is the integral of this n-form over M̃ . In
this section we use the latter meaning.
One can check that if π is a diffeomorphism, then δA(Φπ , V) is the derivative at t = 0 of the n-dimensional surface area of Φt = Φπ + tV. Since we
will not use this fact, we do not prove it here. We need a more complicated
formula to handle the case of non-injective and singular π.
We think of Φ as a minimal surface, and therefore it is natural to expect
that the first variation of surface area is zero. Indeed, this is the case, and
the rest of this section is devoted to a proof of the following key proposition:
Proposition 7.2. δA(Φπ , V) = 0.
The proof consists of two parts. First, we compute the integrand of (1)
at a point x ∈ M̃ . The result is written in terms of derivatives of π and the
coordinate functions {Vα }α∈S of V.
Second, we represent the resulting expression as a differential form in a
suitable manifold and integrate it using Stokes’ formula. While this computation is probably valid for functions of so low regularity as we have, we
do not verify this for every formula. Instead, we perform the computation
assuming that the maps π and V are smooth. Then the general case follows by approximation. Indeed, we do not use any specific properties of our
maps except that Φπ = Φ ◦ π and that π : M̃ → M is a Lipschitz map,
so the computation proves the identity δA(Φπ , V) = 0 for arbitrary smooth
maps π : M̃ → M and V : M̃ → L. The identity then follows for all Lipschitz maps since the integrand of (1) is expressed in terms of the first-order
derivatives.
In addition, note that δA(Φπ , V) is independent of the Riemannian metric
on M̃ , so the fact that it is only piecewise C 0 does not play any role.
Notation. We denote by λ the oriented Riemannian volume form of (M, g).
That is, if y ∈ M and v1 , . . . , vn ∈ Ty M , then λ(v1 , . . . , vn ) is the oriented
volume of the parallelotope spanned by v1 , . . . , vn .
If ξ is an exterior k-form on a vector space X and v ∈ X, then u¬ξ
denotes the (k − 1)-form on X defined by
(v¬ξ)(v1 , . . . , vn−1 ) = ξ(v, v1 , . . . , vn−1 )
14
for all v1 , . . . , vn−1 ∈ X. If ξ is a differential form and v is a vector field, the
notation is applied point-wise.
Point-wise computation. Fix x ∈ M̃ and denote y = π(x) ∈ M . To
avoid cumbersome formulas, we introduce the following temporary notation:
U = U Ty M , W = Wy = dy Φ(Ty M ). We regard W as a subspace of the
Euclidean space Ly with the scalar product h, iy .
Recall that the unit sphere U with the standard normalized volume du is
identified with (S, µy ) via a map ωy : U → S (cf. Definition 6.3). Then we
can “change coordinates” in L by identifying it with L∞ (U ); this way h, iy
becomes the standard scalar product in L2 (U, du).
Lemma 7.3. Let L : Tx M̃ → L be a linear map with coordinate functions
{Lα }α∈S , then
Z
nn
λ(u1 , . . . , un ) lu1 ∧ lu2 ∧ · · · ∧ lun du1 . . . dun ,
(2)
JW (L) =
n! U n
where lu = Lωy (u) .
Proof. Let {e1 , e2 , . . . en } be an orthonormal positively oriented basis in
Ty M . Then
JW (L) = P1 ∧ P2 · · · ∧ Pn ,
(3)
where Pi is a linear function on Tx M̃ defined by
Pi (v) = hL(v), dy Φ(ei )iy .
Indeed, dy Φ is an isometric embedding of Ty M to Ly (cf. Proposition 6.4)
and Pi is a composition of L and the orthogonal projection to the image
of ei . Then by definition of the scalar product in Ly ,
Z
Z
Pi (v) = n Lα (v)dy Φα (ei ) dµy (α) = n Lα (v)hG(y, α), ei i dµy (α).
S
S
(recall that G(y, α) = grad Φα (y)). Using the definition of µy (cf. 6.3) we
rewrite the formula as
Z
Pi (v) = n
lu (v)hu, ei i du.
U
Then (3) takes the form
Z
JW (L) = nn
lu1 ∧ lu2 ∧ · · · ∧ lun hu1 , e1 ihu2 , e2 i . . . hun , en i du1 . . . dun .
Un
Note that if we replaced the basis {ei } by another one obtained by permuting
the vectors e1 , e2 , . . . , en , the same formula holds for positive permutations,
and it acquires a minus sign for negative ones. Adding these formulas for
all permutations of {e1 , e2 , . . . en }, we get
Z
n!JW (L) = nn
lu1 ∧ lu2 ∧ · · · ∧ lun det(hui , ej i)ni,j=1 du1 . . . dun .
Un
15
We complete the proof of the lemma by noting that the determinant of the
matrix (hui , ej i) is just the oriented volume of the parallelotope spanned by
u1 , u2 , . . . , un .
Lemma 7.4. If L = dx Φπ and V : Tx M̃ → L is a linear map with coordinates {Vα }α∈S , then
Z
(4)
δJW (L, V ) = c(n)
vu ∧ π ∗ (u¬λ) du
U
where vu = Vωy (u) and
ative of ) π.
π∗
denotes the pull-back of a form under (the deriv-
Proof. As in Lemma 7.3, define lu = Lωy (u) where {Lα }α∈S are coordinate
functions of L. Then for ξ ∈ Tx M̃ , u ∈ U and α = ωy (u), we have
lu (ξ) = Lα (ξ) = dy Φα (dx π(ξ)) = hG(y, α), dx π(ξ)i = hu, dx π(ξ)i.
Introducing a co-vector u◦ ∈ Ty∗ M by u◦ = hu, ·i, we rewrite this formula as
lu = π ∗ (u◦ ).
d
To compute δJW (L, V ) = dt
J (L + tV ), we plug lu + tvu for lu in (2)
t=0 W
and differentiate it with respect to t. We get
^ Z X
n
nn
k−1
(−1) λ(u) vuk ∧
lui du
δJW (L, V ) =
n! U n
(5)
k=1
i6=k
where u stands for (u1 , . . . , un ) and du for du1 . . . dun . Using the symmetry
of the formula with respect to permuting ui ’s, we rewrite it as
^
Z
Z
n
nn+1
nn+1
(6) δJW (L, V ) =
λ(u) vu1 ∧
vu ∧A(u) du,
lui du =
n! U n
n! U
i=2
where A(u) is an (n − 1)-form on Tx M̃ given by
Z
n−1
^
A(u) =
λ(u, u1 , . . . , un−1 )
lui du1 . . . dun−1 .
U n−1
i=1
¿From (5) we have lui =
π ∗ (u◦i ),
then
A(u) = π ∗ (B(u))
where
Z
B(u) =
U n−1
n−1
^
◦
λ(u, u1 , . . . , un−1 )
ui du1 . . . dun−1 .
i=1
Observe that B(u) depends only on u and the Euclidean structure of Ty M , in
particular, it is equivariant under the action of the orthogonal group. Such
an (n − 1)-form is unique up to a constant factor, and u¬λ is an example
of such a form. Therefore B(u) = c1 (n)u¬λ, A(u) = c1 (n)π ∗ (u¬λ) and the
lemma follows by plugging this into (6).
16
Changing the variable u to α = ωy (u) under the integral in (4), we get
Z
δJW (L, V ) = c(n) Vα ∧ π ∗ (G(y, α)¬λ) dµy (α).
S
(recall that G(y, α) = ωy−1 (α)). This finishes the point-wise computation for
which we needed temporary notation. Substituting the definitions of L, y
and U , we get
Z
π
δJWπ(x) (dx Φ , V ) = c(n) Vα ∧ π ∗ (G(π(x), α)¬λ) dµy (α).
S
Substituting dx V for V (assuming that V is weakly differentiable at x) yields
Z
π
(7) δJWπ(x) (dx Φ , dx V) = c(n) dx Vα ∧ π ∗ (G(π(x), α)¬λ) dµy (α).
S
where {Vα }α∈S are the coordinate functions of V.
Integration of the form. Note that the expression in (7) (as a function
of x) is a differential n-form on M̃ , and δA(Φπ , V) is the integral of this
form over M̃ . We are going to rewrite this as an integral of a differential
(2n − 1)-form over M̃ × S. Define a map P : M̃ × S → M × S by
P (x, α) = (π(x), α),
x ∈ M̃ , α ∈ S.
We need (n−1)-forms σ and σ̃ on M ×S and M̃ ×S to represent integration
over the family of measures µy , y ∈ M . Namely define
σ(y, α) = P2∗ µy (α),
y ∈ M, α ∈ S,
where P2 : M × S → S is the coordinate projection and µy is regarded as
an (n − 1)-form on S. Similarly define
σ̃(x, α) = P̃2∗ µπ(x) (α),
x ∈ M̃ , α ∈ S.
where P̃2 is the coordinate projection M̃ × S → S. Note that σ̃ = P ∗ (σ).
We say that a differential form ξ on M × S represent a family of forms
{ξα }α∈S on M if for every α ∈ S, ξα = ξ|M ×{α} , more precisely, ξα = i∗α (ξ)
where iα : M → M × S is defined by iα (x) = (x, α). One easily checks the
following properties:
1. If forms ξ and η represent families {ξα }α∈S and {ηα }α∈S , then ξ ∧ η
represents {ξα ∧ ηα }α∈S .
2. If a form ξ on M × S represents a family {ξα }α∈S of forms on M , then
the form P ∗ ξ on M̃ × S represents the family {π ∗ ξ} of forms on M̃ .
3. If ξ is an n-form on M̃ × S representing a family {ξα }α∈S , then
Z Z
Z
ξα (x) dµπ(x) (α) dx =
ξ ∧ σ̃.
M̃
M̃ ×S
S
Combining this with (7) we get
Z
Z
(8)
δA(Φπ , V) =
δJWπ(x) (dx Φπ , dx V) dx = c(n)
M̃ ×S
M̃
17
ξ ∧ P ∗ η ∧ σ̃
where ξ is any 1-form on M̃ ×S representing the family {dx Vα }α∈S of 1-forms
on M̃ , η is an (n − 1)-form on M × S representing the family {Gα ¬λ}α∈S
of (n − 1)-forms on M . Here Gα is a vector field on M defined by Gα (x) =
G(x, α).
We have to specify ξ and η in (8). First define ξ = dF where the function
F : M̃ × S → R is given by
F (x, α) = Vα (x).
(9)
Obviously ξ = dF represents the family {dx Vα }α∈S .
To define η, introduce a vector field γ on M × S so that for every (y, α) ∈
M × S the projection of the vector γ(y, α) to M equals Gα (y) and the
projection to S is zero. Let λ0 denote the n-form on M × S computing the
oriented Riemannian volume of the projection to M . Note that λ0 is the
pull-back of λ under the coordinate projection M × S → M . Now define
η = γ¬λ0 .
The definitions imply that η represents the family {Gα ¬λ}α∈S .
Plugging ξ = dF into (8), we get
Z
δA(Φπ , V) = c(n)
dF ∧ P ∗ η ∧ σ̃.
M̃ ×S
Using the identity σ̃ =
(10)
P ∗ σ,
we rewrite this as follows:
Z
δA(Φπ , V) = c(n)
dF ∧ P ∗ (η ∧ σ).
M̃ ×S
Recall that G : U T M → M × S is a diffeomorphism, and the measure
dµy dy on M × S (where du is the Riemannian volume on M ) is the pullback of the Liouville measure on U T M under G. Denote by µ the differential
(2n − 1)-form on M × S corresponding to this measure. Then
µ = λ0 ∧ σ
by the definitions of λ0 and σ. Observe that γ¬σ = 0 since γ is tangent to
the fibers M × {α} and these fibers annulate σ. Hence
η ∧ σ = (γ¬λ0 ) ∧ σ = γ¬(λ0 ∧ σ) = 㬵.
Then (10) takes the form
(11)
π
Z
δA(Φ , V) = c(n)
dF ∧ P ∗ (㬵).
M̃ ×S
For every α ∈ S, the vector field γ on a M × {α} projects to the vector
field Gα = grad Φα on M . The trajectories of Gα are geodesics since Φα is
a distance function. Hence the flow on M × S generated by γ is mapped
by G to the geodesic flow on U T M . Since the geodesic flow preserves the
Liouville measure, the flow generated by γ preserves µ. This implies that
㬵 is a closed form. Then P ∗ (㬵) is closed: d(P ∗ (㬵)) = 0. Therefore
dF ∧ P ∗ (㬵) = d(F · P ∗ (㬵))
18
Then from (11),
π
Z
Z
∗
δA(Φ , V) = c(n)
d(F · P (㬵)) = c(n)
M̃ ×S
F · P ∗ (㬵))
∂ M̃ ×S
by Stokes’ formula. The last integral is zero since F vanishes on the boundary of M̃ × S (cf. (9)). This finishes the proof of Proposition 7.2.
8. An Estimate on δJ
Let H be a (possibly infinite-dimensional) Euclidean space and X an
oriented Euclidean n-space. For a linear map L : X → H we denote by
J(L) the (nonnegative) Jacobian of L.
Let W be an oriented n-dimensional subspace of H. We use the notation
JW (L) and δJW (L, V ) from Definition 7.1 for linear maps L, V : X → H.
Proposition 8.1. There exists a constant ε = ε(n) > 0 such that the following holds. In the above notation, if L(X) ⊂ W and V (X) ⊂ Q where
Q ⊂ H is a codimension n linear subspace and Q is ε-orthogonal to W (cf.
Definition 6.5), then
(12)
J(L + V ) ≥ JW (L) + δJW (L, V ),
and the equality implies that either V = 0 or both L and L+V are degenerate
(have ranks less than n), and in either case J(L + V ) = JW (L).
Proof. The images of maps L, V and L + V are contained in the subspace
W + L(X) of dimension at most 2n. Therefore it suffices to prove the
proposition in the case when dim H = 2n. Then dim W = dim Q = n.
Introduce a family of linear maps Lt : X → H, t ∈ [0, 1] by Lt = L + t · V .
Then by definition,
d δJW (L, V ) = JW (Lt ).
dt t=0
We will show that
(13)
J(Lt ) ≥ JW (L) + t · δJW (L, V )
for all t ≥ 0; then (12) follows by substituting t = 1.
If α ∈ Λn (H) is a decomposable n-vector α = v1 ∧ v2 ∧ · · · ∧ vn , we denote
by kαk the n-volume of the parallelotope spanned by v1 , v2 , . . . , vn . Note
that the scalar product h, i in H canonically determines a scalar product in
Λn (H). We also denote this scalar product by h, i. Then k · k is a Euclidean
norm on Λn (H) corresponding to this scalar product.
Denote Λk = Λk (W ) ∧ Λn−k (Q). The assumption that Q and W are
almost orthogonal implies that Λi and Λj (i 6= j) are almost orthogonal.
Namely, if ξ ∈ Λi and η ∈ Λj (i 6= j) then
(14)
hξ, ηi ≤ ε1 kξk kηk
for some ε1 = ε1 (ε, n), ε1 → 0 as ε → 0.
19
Let α(t) ∈ Λn (H) denote the image of the unit positively oriented n-vector
in Λn (X) ' R under (Lt )∗ . In other words,
α(t) = Lt (e1 ) ∧ Lt (e2 ) ∧ · · · ∧ Lt (en )
where e1 , e2 , . . . , en is a positive orthonormal basis of X. Then J(Lt ) =
kα(t)k. Obviously α(t) is a polynomial of the form
n
X
(15)
α(t) =
αi ti ,
i=0
where αi ∈ Λi .
Lemma 8.2. Assuming that ε is sufficiently small, there exists a constant
c(n) such that
(16)
kα0 k kαk k ≤ c(n) kα1 k kαk−1 k
where αi are defined by (15).
Proof. Since Q and W are ε-orthogonal, applying a linear transformation
making them orthogonal changes all norms in the exterior algebra by factors
close to 1. Thus we can assume that Q and W are orthogonal, and identify
H = W ⊕ Q with Rn × Rn .
If L0 is degenerate then the left-hand side of (16) is zero, and the inequality is obvious. Otherwise we can choose a basis in X so that the
matrix {Lij , i = 1, 2 . . . 2n, j = 1, 2 . . . n} of L0 consists of two blocks: the
identity matrix {Lij , i = 1, 2 . . . n, j = 1, 2 . . . n} (corresponding to the projection to W ) and zero matrix {Lij , i = n + 1, n + 2 . . . 2n, j = 1, 2 . . . n}
(corresponding to the projection to Q). Then the first block of Lt remains
the identity matrix for all t (by the definition of the family {Lt }, and the
second block is tB, where B is some (fixed) matrix. Even though the norms
on exterior powers depend on the choice of a basis, both parts of (16) get
multiplied by the same constant, and hence changing coordinates in X is an
admissible procedure.
Note that kαk k2 is the sum of the squares of all n×n-minors of (the matrix
of ) L1 such that exactly k rows are chosen in the lower half of the matrix
(that is, in B). Since the upper-half of Lt is the identity matrix, every such
minor is equal to a k × k-minor of B. Hence kαk k2 is the binomial coefficient
times the sum of the squares of all k × k-minors of B.
In our coordinates, α0 = 1. Since every k × k-minor is a sum of products
of (k − 1) × (k − 1)-minors and 1 × 1-minors, the lemma follows.
Let σ denote the unit positively oriented n-vector in Λn W ' R. Note
that JW (β) = hσ, βi for every β ∈ Λn (H). Hence δJW (L, V ) = hα1 , σi and
JW (L0 ) = hα0 , σi. Thus (13) takes the form
kα(t)k ≥ hα0 , σi + thα1 , σi,
or, after squaring (note that the left-hand side is nonnegative),
kα(t)k2 ≥ hα0 , σi2 + 2thα0 , σihα1 , σi + t2 hα1 , σi2 .
20
Since α0 is proportional to σ and kσk = 1, we have |hα0 , σi| = kα0 k and
hα0 , σihα1 , σi = hα0 , α1 i. Thus the desired inequality takes the form
kα(t)k2 ≥ kα0 k2 + 2thα0 , α1 i + t2 hα1 , σi2 .
We will actually prove the following stronger inequality:
1
(17)
kα(t)k2 ≥ kα0 k2 + 2thα0 , α1 i + t2 hα1 , σi2 + kα(t) − α0 k2 .
10
1
2
The additional term 10 kα(t) − α0 k in the right-hand side of this inequality
will help us to analyze the equality case in (13).
Denote β(t) = t2 α2 · · · + tn αn , then α(t) = α0 + tα1 + β(t) and
kα(t)k2 = kα0 k2 + 2thα0 , α1 i + t2 kα1 k2 + 2hα0 , β(t)i + 2thα1 , β(t)i + kβ(t)k2 .
Since α1 ∈ Λ1 is ε1 -orthogonal to σ ∈ Λ0 , we have kα1 k2 ≥ 10hα1 , σi2 , so it
suffices to prove that
9 2
1
t kα1 k2 + 2hα0 , β(t)i + 2thα1 , β(t)i + kβ(t)k2 ≥ kα(t) − α0 k2 .
10
10
Since α1 is ε1 -orthogonal to each Λi , i > 1, which in √
their turn are also
almost orthogonal, one can easily see that α1 is, say, 2 nε1 -orthogonal to
β(t) ∈ Λ2 ⊕ · · · ⊕ Λn (provided that ε1 is small enough). Then we have
1 2
1
t kα1 k2 + 2thα1 , β(t)i + kβ(t)k2 ≥ 0.
10
10
It remains to prove that
8 2
9
1
t kα1 k2 + 2hα0 , β(t)i + kβ(t)k2 ≥ kα(t) − α0 k2 .
10
10
10
Observe that
1
1
2 2
kα(t) − α0 k2 = ktα1 + β(t)k2 ≤
t kα1 k2 + kβ(t)k2 ,
10
10
10
hence it suffices to prove that
6 2
7
(18)
t kα1 k2 + 2hα0 , β(t)i + kβ(t)k2 ≥ 0.
10
10
Combining the triangle inequality with (14) and (16), we get
|hα0 , β(t)i| ≤
n
X
i
|hα0 , t αi i| ≤ ε1
i=2
n
X
i
t kα0 kkαi k ≤ ε1 c(n)
i=2
We may assume that ε1 c(n) <
1
10 .
n
X
ti kα1 kkαi−1 k.
i=2
Then, separating the first term, we get
n
|hα0 , β(t)i| ≤
X
1 2
t kα1 k2 + ε1 c(n)
ti kα1 kkαi−1 k.
10
i=3
Using the above inequality, one sees that, to prove (18) it suffices to show
that
n
X
4 2
7
(19)
t kα1 k2 − 2ε1 c(n)
ti kα1 kkαi−1 k + kβ(t)k2 ≥ 0.
10
10
i=3
21
Recall that β(t) =
hence
Pn
i
i=2 t αi ,
and the terms ti αi are mutually ε1 -orthogonal,
n
kβ(t)k2 ≥
3 X 2i
t kαi k2
4
i=2
provided that ε1 small enough (and
(19) follows from
3
4
is just a number smaller than 1). Then
n
n−1
(20)
X
4 2
4 X 2i
t kα1 k2 − 2ε1 c(n)
t kαi k2 ≥ 0.
ti+1 kα1 kkαi k +
10
10
i=2
i=2
We assume that ε is so small that ε1 c(n) <
1
10n .
Then
1
1 2
t kα1 k2 − 2ε1 c(n)ti+1 kα1 kkαi k + t2i kαi k2 ≥ 0,
10n
10
for all i = 2, 3, . . . , n − 1, and the desired inequality (20) follows by adding
them.
Now let us consider the equality case in (12), or, equivalently, in (13) for
t = 1. Since we proved a stronger inequality (17), the equality implies that
kα(1) − α0 k = 0. Hence the images of L and L1 = L + V either coincide or
degenerate (of dimension less than n). Furthermore, since the image of L is
almost orthogonal to the image of L, this implies that V = 0 unless L has
rank smaller than n, in which case V has rank smaller than n as well. Since
α(1) − α0 = α1 + α2 + · · · + αn and the terms
L αi belong to the respective
n
components Λi of the direct sum Λ (H) =
Λi , it follows that αi = 0 for
all i ≥ 1. Then δJW (L, V ) = hα1 , σi = 0, hence J(L + V ) = JW (L).
9. Proof of Theorem 2
Let x ∈ M̃ be such that Φ̃ is weakly differentiable at x. Consider X =
Tx M̃ , H = Lπ(x) , L = dx Φπ : X → H, V = dx V : X → H (cf. Notation 6.7)
and W = Wπ(x) (cf. Definition 7.1). Note that L(X) ⊂ W . By Proposition
6.6, L(V ) ⊂ Q where Q is ε-orthogonal to W for a small ε. Then Proposition
8.1 applies, and we have
(21)
Jx (Φ̃) ≥ JWπ(x) (dx Φπ ) + δJWπ(x) (dx Φπ , dx V).
By means of integration we get
Z
Area(Φ̃) ≥
JWπ(x) (dx Φ̃π ) dx + δA(Φπ , V).
M̃
(cf. Definitions 6.8 and 7.1). By Proposition 7.2, the last term is zero, thus
Z
(22)
Area(Φ̃) ≥
JWπ(x) (dx Φπ ).
M̃
Recall that Φπ = Φ ◦ π and hence dx Φπ = dπ(x) Φ ◦ dx π. By Definition 7.1
and Lemma 6.9, dπ(x) Φ is an orientation-preserving isometry from Tπ(x) M to
Wπ(x) . Hence the integrand JWπ(x) (dx Φπ ) of (22) is nothing but the signed
22
Jacobian of the map π : M̃ → M at x. Then the right-hand part of (22)
equals the volume of (M, g), thus
Area(Φ̃) ≥ vol(M, g).
By Lemma 6.9 we have Area(Φ̃) ≤ vol(M̃ , g̃), and the inequality part of the
theorem follows.
To analyze the equality case, note that all the above inequalities have to
turn into equalities almost everywhere on M̃ . The equality part of Lemma
6.9 implies that Jx (Φ̃) = 1 for a.e. x ∈ M̃ . Then by Proposition 8.1, the
equality in (21) implies that
JWπ(x) (dx Φπ ) = Jx (Φ̃) = 1
for a.e. x ∈ M̃ . Hence by the equality case of Proposition 8.1, we conclude
that dx V = 0 (that is, the tangent spaces to the images of Φ and Φ̃ at
corresponding points are parallel). Again observe that JWπ(x) (dx Φπ ) equals
the signed Jacobian of π at x, and thus we get that dx π is an orientationpreserving linear isometry from Tx M̃ to Tπ(x) M for almost all x ∈ M̃ .
Now the Theorem follows from the following lemma (compare with Sublemma for Theorem 1 iof [4] and Appendix C of [7]) :
Lemma 9.1. Let M̃ be a piece-wise C 0 Riemannian manifold and M a
smooth Riemannian manifold and vol(M̃ ) = vol(M ). Let π : M̃ → M be a
surjective Lipschitz map such that the differential dx π is a linear isometry
for almost all x, and π(∂ M̃ ) ⊂ ∂M . Then π is an isometry.
Proof. Since dx π is a linear isometry for almost all x ∈ M̃ , π is a Lipschitz-1
map. Hence it is volume-nonexpanding. Then the assumption vol(M̃ ) =
vol(M ) = vol(π(M̃ )) implies that π is volume-preserving: vol(π(U )) =
vol(U ) for every measurable set U ⊂ M̃ .
Recall that M̃ is triangulated into n-dimensional simplices with C 0 Riemannian metrics. Let Σ the union of ∂ M̃ and the (n − 2)-skeleton of the
triangulation.
For an x ∈ M̃ , we denote by Cx the tangent cone of M̃ at x. By definition,
it is a length space identified with the vector space Tx M̃ (or half-space if
x ∈ ∂ M̃ ) and split into a number of polyhedral cones corresponding to
simplices adjacent to x. Each cone carries a flat metric defined by the
Riemannian tensor of the corresponding simplex at x, and the whole metric
of Cx is obtained by gluing these Euclidean metrics together in the usual
length metric sense.
It is easy to see that the volume of a small metric ball centered at x ∈ M̃
is approximately equal to that of a similar ball in Cx . More precisely,
vol(Bε (x)) = vol(B)εn + o(εn ),
ε → 0,
where B is a unit metric ball in Cx centered at the origin. (Note that B
may be larger than the union of balls in the polyhedral cones that form Cx
since non-isometric gluing can decrease distances). If x ∈ M̃ \ Σ, then the
23
tangent cone is a Euclidean space or the result of gluing of two half-spaces
along a linear map between their boundaries. Hence
vol(Bε (x)) ≥ ωn εn + o(εn ),
ε → 0,
where ωn is the volume of the standard Euclidean n-ball,
We prove the lemma in a number of steps.
1. The map π1 := π|M̃ \Σ is injective and its image is contained in M \∂M .
Suppose that π(x) = π(y) for some x, y ∈ M̃ \ Σ, x 6= y. For a sufficiently
small ε > 0, the balls Bε (x) and Bε (y) are disjoint and contained in M̃ \ Σ.
Since Cx is either a Euclidean space or a union of two half-spaces, we have
vol(Bε (x)) ≥ ωn εn + o(εn ),
ε → 0,
and similarly for y, thus
vol(Bε (x) ∪ Bε (y)) ≥ 2ωn εn + o(εn ),
ε → 0.
Since π is Lipschitz-1, the images of balls Bε (x) and Bε (y) are contained in
the ε-ball centered at π(x) = π(y). On the other hand,
vol(Bε (π(x))) = ωn εn + o(εn ) < vol(Bε (x) ∪ Bε (y))
contrary to the fact that π is volume-preserving. Thus π1 is injective.
The second statement follows similarly: if x ∈ M̃ \ Σ and π(x) ∈ ∂M ,
then
1
vol(Bε (π(x))) = ωn εn + o(εn ) < vol(Bε (x)),
2
a contradiction.
2. The metrics of the adjacent simplices of the triangulation of M̃ agree
on the (n − 1)-dimensional faces.
Let x ∈ M̃ \ Σ. The tangent cone Cx is obtained by gluing together two
Euclidean half-spaces H1 and H2 . We have to show that the metrics of H1
and H2 agree on their common hyperplane. Suppose the contrary. Then
some points are closer to the origin in Cx than they would be in the disjoint
union of H1 and H2 . Hence the unit ball in Cx is strictly larger that the
union of two Euclidean half-balls in H1 and H2 , therefore the volume of the
ball is greater than ωn . Thus
vol(Bε (x)) = Cωn εn + o(εn )
for some C > 1. This leads to a contradiction as in Step 1.
3. The map π1 = π|M̃ \Σ is a locally bi-Lipschitz homeomorphism onto an
open subset of M \ ∂M .
Since M̃ \Σ and M \∂M are n-dimensional manifolds without boundaries,
by the Brouwer Invariance of Domain Theorem ([3]) the injectivity implies
that π1 is an open map, hence its inverse π1−1 is continuous.
Since the metrics agree on the (n − 1)-dimensional faces of M̃ , we may
regard M̃ \ Σ as a manifold (with some differential structure) with a C 0
24
Riemannian metric. Note that the continuity of metric coefficients implies
that the relation
vol(Bε (x)) = ωn εn + o(εn ),
ε → 0,
is uniform in x on any compact subset of M̃ \ Σ, and similarly in M \ ∂M .
Fix an x ∈ M̃ \ Σ, let y be sufficiently close to x, and suppose that ε :=
|π(x)π(y)| < 12 |xy|. Then the balls Bε (x) and Bε (y) are disjoint, therefore
vol(π(Bε (x) ∪ Bε (y))) = vol(Bε (x) ∪ Bε (y)) = 2ωn εn + o(εn ).
On the other hand, π(Bε (x) ∪ Bε (y)) ⊂ Bε (π(x)) ∪ Bε (π(y)) since π is
Lipschitz-1, but the balls Bε (π(x)) and Bε (π(y)) contain a ball of radius
ε/2 in their intersection, therefore
vol(Bε (π(x)) ∪ Bε (π(y))) ≤ 2 − 1/2n ωn εn + o(εn ) < vol(π(Bε (x) ∪ Bε (y)))
if ε is small enough. This contradiction shows that |π(x)π(y)| ≥ 21 |xy| if y
is sufficiently close to x. It follows that π1−1 is locally Lipschitz at π(x).
4. π is an isometry.
First observe that π1 is an isometry of length spaces M̃ \ Σ and π(M̃ \ Σ).
Indeed, since π1−1 is Lipschitz, it is differentiable a.e., and its differential,
wherever defined, is the inverse of that of π. Then dy (π1−1 ) is a linear
isometry for almost all y ∈ π(M̃ \ Σ). It follows that π1−1 is Lipschitz-1
(with respect to the induced length distances). Since both π1 and π1−1 are
Lipschitz-1, π1 is an isometry (of induced length metrics).
It remains to show that the induced length metrics on M̃ \Σ and π(M̃ \Σ)
coincide with the restrictions of the ambient metrics of M̃ and M . It follows
from the fact that the sets Σ in M̃ and π(Σ) are “small”: each of them
consists of a subset of a boundary and a set of Hausdorff dimension at most
n − 2. Every piecewise curve can be perturbed so as to avoid such a set
while almost preserving the length, so removing these sets does not change
the length distances.
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Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
E-mail address: [email protected]
Steklov Math. Institute, St. Petersburg, Russia
E-mail address: [email protected]
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